Vector field computations in Cli ord's geometric ... - Semantic Scholar

E. Hitzer, R. Bujack, G. Scheuermann, Vector field computations in Clifford's geometric algebra, Proc. of the Third SICE Symposium on Computational Intelligence, August 30, 2013, Osaka University, Osaka, pp. 91-95.

Vector eld computations in Cliord's geometric algebra * Eckhard Hitzer (International Christian University, Japan), Roxana Bujack and Gerik Scheuermann (Leipzig University, Germany)

Abstract Exactly 125 years ago G. Peano introduced the modern concept of vectors

in his 1888 book "Geometric Calculus - According to the Ausdehnungslehre (Theory of Extension) of H. Grassmann". Unknown to Peano, the young British mathematician W. K. Cliord (1846-1879) in his 1878 work "Applications of Grassmann's Extensive Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern concept of geometric algebras, including the measurement of lengths (areas and volumes) and angles (between arbitrary subspaces). This leads currently to new ideal methods for vector eld computations in geometric algebra, of which several recent exemplary results will be introduced.

Keywords: Vector eld, Cliord's geometric algebra, Geometric calculus

1 Introduction The descriptions of L. Kannenberg's rst complete translation [2] of G. Peano's Calcolo Geometrico says:

Calcolo Geometrico, G. Peano's rst pub-

lication in mathematical logic, is a model of expository writing, with a signicant impact on 20th century mathematics. . . . In Chapter IX, with the innocent-sounding title "Transformations of a linear system," one nds the crown jewel of the book: Peano's axiom system for a vector space, the rst-ever presentation of a set of such axioms. The very wording of the axioms (which Peano calls "denitions") has a remarkably modern ring, almost like a modern introduction to linear algebra. Peano also presents the basic calculus of set operation, introducing the notation for 'intersection,' 'union,' and 'element of,' many years before it was accepted. Despite its uniqueness, Calcolo Geometrico has been strangely neglected by historians of mathematics, and even by scholars of Peano. The mathematical biography of G. Peano at the University of St. Andrews [3] writes about Calcolo Geometrico : In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic. . . . A more signicant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself. This book contains the rst denition of a

vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano. In the preface G. Peano himself writes in February 1888 about his expectations for his work [2]: . . . I will be satised with my work in writing this book (which would be the only recompense I could expect), if it serves to disclose among mathematicians some of the ideas of Grassmann [1]. It is however my opinion that, before long, this geometric calculus, or something analogous, will be substituted for the methods actually in use in higher education. It is indeed true that the study of this calculus, as with that of every science, requires time; but I do not believe that it exceeds that necessary for the study of, e.g., the fundamentals of analytic geometry; and then the student will nd himself in possession of a method which comprehends that of analytic geometry as a particular case, but which is much more powerful, and which lends itself in a marvelous way to the study of geometric applications of innitesimal calculus, of mechanics, and of graphic statics; indeed, some part of such sciences are already observed to have taken possessions of that calculus. . . . Grassmann's work was the source for W.K. Cliord [4, 5] in England to introduce the modern concept of geometric algebras, which includes the measurement of lengths (areas and volumes) and angles (between arbitrary subspaces). He wrote [4]: . . . I propose to communicate in a brief form

some applications of Grassmann's theory . . . I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast inuence upon the future of mathematical science. ... A recent review (covering works until early 2012) of modern applications of Cliord's geometric algebra can be found in [7]. We will therefore concentrate on more recent advances. Regarding recent progress, we want to report in Section 2 about the detection of outer and total rotations in two-dimensional and three-dimensional vector elds using iterative geometric correlation [9, 10, 11]. Another area of major progress is based on the in depth study of square roots of −1 in Clifford's geometric algebras [12]. This has lead to new research (see Section 3) in quaternion and Cliord Fourier transformations [16, 17, 18, 19, 20, 22, 23, 24]. Next we explain (Section 4) about the establishment of the algebraic foundations of split hypercomplex nonlinear adaptive ltering [26]. And nally (Section 5) we show how even in material science, Cliord's geometric algebra allows to nd new geometric descriptions for fundamental symmetry properties [27].

2 Progress in vector eld detection Correlation is a common technique for the detection of shifts. Its generalization to the multidimensional geometric correlation in Cliord algebras additionally contains information with respect to rotational misalignment. It has proven a useful tool for the registration of vector elds that dier by an outer rotation. In [11] we have recently proved that applying the geometric correlation iteratively has the potential to detect the total rotational misalignment for linear two-dimensional vector elds. We have further analyzed its eect on general analytic vector elds and showed how the rotation can be calculated from their power series expansions. So far the exact correction of a three-dimensional outer rotation could only be achieved in certain special cases. In [10] we further prove that applying the geometric correlation iteratively can detect the outer rotational misalignment even for arbitrary three-dimensional vector elds. Thus, we developed a foundation applicable for image registration, color image processing and pattern matching. Based on the theoretical work we have established a new algorithm and tested it on several principle examples. In [9] we further present the explicit iterative algorithm, and analyze its eciency for detecting the

rotational misalignment in the color space of a color image. The experiments suggest a method for the acceleration of the algorithm, which has now been practically tested with great success.

3 Progress in quaternion and Cliord Fourier transforms Vector elds can be embedded with great advantage in a Cliord algebra, which contains the corresponding vector space as a subset. Due to the availability of new types of Fourier transformations in the embedding Cliord algebra, new methods can be established for vector eld processing. A major new type of Cliord Fourier transformation relies on the detailed study [12] of square roots of −1 in Cliord algebras Cl(p, q), n = p + q . It is well known that real Cliord (geometric) algebra oers a real geometric interpretation for square roots of −1 in the form of blades that square to minus one. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [14] on the biquaternion roots of −1, abandoning the restriction to blades. Biquaternions are isomorphic to the Cliord (geometric) algebra Cl(3, 0) of R3 . Further research on general algebras Cl(p, q) has explicitly derived the geometric roots of −1 for p + q ≤ 4 [13]. The new research [12] abandons this dimension limit and uses the Cliord algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of −1 found in the dierent types of Cliord algebras, depending on the type of associated ring (R, H, R2 , H2 , or C). This allows to establish explicit computer generated tables of representative square roots of −1 for all Cliord algebras with n = 5, 7, and s = 3 (mod 4) with the associated ring C. This includes, e.g., Cl(0, 5) important in Cliord analysis, and Cl(4, 1) which in applications [7] is at the foundation of conformal geometric algebra. All these roots of −1 are immediately useful in the construction of new types of geometric Cliord Fourier transformations (CFT). Basically in the kernel of the complex Fourier transform the imaginary unit j in C (complex numbers) is replaced by a real geometric multivector square root of −1 in Cl(p, q). The recent (onesided) CFT [18] thus obtained generalizes previously known and applied CFTs [15], which replaced j in C only by blades (usually pseudoscalars) squaring to −1. A major advantage of real Cliord algebra CFTs is their completely real geometric interpretation. Established have been so far (left and right) linearity of the CFT for constant multivector coecients in Cl(p, q), translation (x-shift) and modulation (ω -shift) properties, and signal dilations. The

new CFTs have an inversion theorem. Moreover, they have been applied to vector dierentials, partial derivatives, vector derivatives and spatial moments of signals. Plancherel and Parseval identities as well as a general convolution theorem have been derived. This research has subsequently been extended to general two-sided CFTs [19], and their properties (from linearity to convolution) have been studied too. Two general multivector square roots ∈ Cl(p, q) of −1 are used both to split multivector signals, and to construct the left and right CFT kernel factors. The classical Fourier Mellin transform [21], which transforms functions f representing, e.g., a gray level image dened over a compact set of R2 has recently been generalized [20] to Hamilton's quaternions. Note that quaternions are isomorphic to Cl(0, 2) and to the even subalgebra Cl+ (3, 0). The quaternionic Fourier Mellin transform (QFMT) applies to functions f : R2 → H, for which |f | is summable over R∗+ × S1 under the measure dθ dr r . R∗+ is the multiplicative group of positive and nonzero real numbers. The properties of the QFMT have been investigated, similar to the investigation of the quaternionic Fourier Transform (QFT) in [8]. The next step of generalization achieved in [22] for the complex Fourier-Mellin transform is to Cliord algebra valued signal functions over the domain Rp,q taking values in Cl(p, q), n = p + q = 2. The two-sided quaternionic Fourier transformation (QFT) was introduced and applied in [25] for the analysis of 2D linear time-invariant partialdierential systems. In further theoretical investigations [8] a special split of quaternions was introduced, then called ±split. In the most recent research [23] this split has been analyzed further, and interpreted geometrically as an orthogonal 2D planes split (OPS), and generalized to a freely steerable split of H into two orthogonal 2D analysis planes. The new general form of the OPS split allows to nd new geometric interpretations for the action of the QFT on the signal. The second major new result is a variety of new steerable forms of the QFT, their geometric interpretation, and for each form, OPS split theorems, which allow fast and ecient numerical implementation with standard FFT software. The increasing demand for Fourier transforms on geometric algebras (CFTs) has resulted in an increasing variety of new transforms. In [16] we therefore introduced one single straight forward denition of a general geometric Fourier transform covering most versions in the literature (up to 2011/2012). We showed which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we can provide guidelines for the target-oriented design of yet unconsidered transforms that fulll requirements in a specic application context. Furthermore, the standard theorems do not

need to be shown in a slightly dierent form every time a new geometric Fourier transform (CFT) is developed since they are proved here once and for all. In further research [17] we extended these results by a general CFT convolution theorem.

4 Progress in hypercomplex nonlinear adaptive ltering A split hypercomplex learning algorithm for the training of nonlinear nite impulse response adaptive lters for the processing of hypercomplex signals of any dimension has recently been proposed in [26]. This includes possible applications to vector signals of any dimension. The derivation of the algorithm strictly took into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have previously been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions it became possible to predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms has been rigorously analyzed.

5 Progress in geometric symmetry description In the eld of material science, recent research work [27] explains how, following the representation of 3D crystallographic space groups in Cliord's geometric algebra [28], has made it possible to similarly represent all 162 so called subperiodic groups of crystallography in Cliord's geometric algebra. A new compact geometric algebra group representation symbol has thus been constructed, which allows to read o the complete set of geometric algebra generators. For clarity moreover the chosen generators have been stated explicitly. The subperiodic group symbols are based on the representation of point groups in geometric algebra by versors (Cliord products of invertible vectors). This is yet another indication, that in many elds the proper way of multiplying vectors is Cliord's associative and invertible geometric product. This way of handling vectors brings many simplications, new geometric understanding and opens up new avenues of research and development. The current paper mainly reviews our own recent work, which is but a small part of the recent new developments in the eld. Further researches and new information can be found in [29] and [30], etc.

Acknowledgment E.H. wants to acknowledge God [31]: In the beginning God created the heavens and the earth. . . . And God said, "Let there be light," and there was light. In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through him all things were made; without him nothing was made that has been made. E.H. further wants to thank his beloved family, as well as T. Nitta and Y. Kuroe.

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E. Hitzer, R. Bujack, G. Scheuermann, Vector field computations in Clifford's geometric algebra, Proc. of the Third SICE Symposium on Computational Intelligence, August 30, 2013, Osaka University, Osaka, pp. 91-95.